Let $B_{t}(\omega)$ be a standard Brownian motion on $(\Omega,\mathcal{F},\mathbb{P})$.
I read in a footnote recently that almost surely the quadratic variation $$(a)\;\;\;\;[B,B](T)=\sup_{\Pi_{[0,T]}}\sum_{n}(B_{t_{j}}(\omega)-B_{t_{j-1}}(\omega))^{2}=+\infty,$$ yet $$(b)\;\;\;\;[B,B](T)\lim_{\Pi_{[0,T]}\to0}\sum_{n}(B_{t_{j}}(\omega)-B_{t_{j-1}}(\omega))^{2}\to T$$ in probability.
If we replace the $\sup$ in (a) with $\lim$ as in (b), does the pathwise computation result in $T$?
Also, we define (with some assumptions not stated here) the Ito integral of $\Delta_{t}(\omega)$ as $$\lim\sum_{n}\Delta_{t_{j-1}}(\omega)(B_{t_{j-1}}(\omega)-B_{t_{j}}(\omega)),$$ where the limit converges in probability to a finite quantity we term the Ito integral and denote by $\int_{0}^{T}\Delta_{t}(\omega)\;dB_{t}(\omega).$
By the same token, this quantity is not well-defined if we fix a realization $\omega$ and take the above limit (as a numerical sequence). This is evidently analogous to the dichotomy between (a) and (b).
Is there a good resources that touches on this? I have been going through my study of stochastic calculus/SDE's taking for granted that the whole theory is based on convergence in probability, which is actually very weird when you think about the traditional theories of integration which are based on convergence of concrete numerical sequences.
This is an edit based on Saz's answer for my own personal benefit.
Consider a stochastic process $f(\omega,t):\Omega\times[0,\infty)\to\mathbb{R}$.
Let $\{\Pi_{N}^{T}\}$ be a collection of a sequence of partitions of $[0,T]$ such that $||\Pi^{T}_{N}||\to0$ as $N\to\infty$ for every $T\geq0$.
Define the sequence of partial sum operators $\{S_{N}\}_{N\geq0}$ by $$S_{N}(f)(\omega, T)=\sum_{j=1}^{N}f(\omega,t_{j-1})(B(\omega,t_{j})-B(\omega,t_{j-1}))$$ and put $$\int_{0}^{T}f(\omega,t)\;dB(\omega,t):=\lim_{N\to\infty}S_{N}(f)(\omega,T).$$
We'd like to understand in what sense $S_{N}(f)$ converges as $N\to\infty$ for adapted and square integrable processes $f(\omega,t)$, and we have the following facts:
- If $f\in L^{p}(\Omega\times\mathbb{R}^{+})$ for $p\leq2$, then $S_{N}(f)$ converges in the $L^{p}$ topology.
- Because of the $L^{p}$ convergence, $S_{N}(f)$ converges in probability on $(\Omega\times\mathbb{R}^{+}, \sigma(\mathcal{F}\times\mathcal{B}), d\mathbb{P}\times dx)$ (and of course $\Omega$ for $T$ fixed as is usually shown).
- For fixed $(\omega,T)$, $S_{N}(f)$ may or may not converge depending on how the partitions were constructed; and in any event, the convergence is not consistent as illustrated in Saz's answer, and therefore we cannot take the pointwise limit as a definition of the integral we are trying to construct.
The main point is that if $$\int_{0}^{T}f(t)\;dB(t)$$ exists in the Lebesgue-Stieltjes sense for every continuous $f$, then necessarily $B(t)$ is of finite variation. If we have a collection of functions of unbounded variation $\{B(\omega,t)\}$, then we cannot define in any reasonable sense the above integral for each $B(\omega,t)$ (i.e. pathwise).
However (and this is actually a rather amazing fact), if we endow the collection of integrators with the structure of a probability space such that the collection $B(\omega,t)$ is a martingale, etc., then we can achieve a limited convergence as detailed above.
For any $\omega \in \Omega$, we can construct a sequence $(\Pi_n)_n=(\Pi_n(\omega))_n$ of partitions of $[0,T]$ such that the mesh size $|\Pi_n|$ tends to $0$ as $n \to \infty$ and
$$\lim_{n \to \infty} \sum_{t_j \in \Pi_n} (B_{t_j}(\omega)-B_{t_{j-1}}(\omega))^2 = \infty.$$
This means in particular that
$$[B,B](T)(\omega) := \sup_{\Pi} \sum_{t_j \in \Pi} (B_{t_j}(\omega)-B_{t_{j-1}}(\omega))^2 = \infty.$$
On the other hand, one can show that there always exists a sequence of partitions $(\Gamma_n)$ of $[0,T]$ such that
$$\lim_{n \to \infty} \sum_{t_j \in \Gamma_n} (B_{t_j}(\omega)-B_{t_{j-1}}(\omega))^2 = T.$$
Consequently, the limit
$$\lim_{|\Pi| \to 0} \sum_{t_j \in \Pi} (B_{t_j}(\omega)-B_{t_{j-1}}(\omega))^2 \tag{1}$$
does not exist.
All the above-mentioned results hold for partitions depending on $\omega$. If we do not allow dependence on $\omega$, then the results are totally different: It is a well-known that the quadratic variation
$$\sum_{t_j \in \Pi_n} (B_{t_j}(\omega)-B_{t_{j-1}}(\omega))^2$$
converges in $L^2(\mathbb{P})$ to $T$ for any sequence of partitions $(\Pi_n)$ of $[0,T]$ such that $|\Pi_n| \to 0$. As $L^2$-convergence implies convergence in probability, we find in particular that
$$\sum_{t_j \in \Pi} (B_{t_j}(\omega)-B_{t_{j-1}}(\omega))^2 \stackrel{\mathbb{P}}{\to} T$$
as $|\Pi| \to 0$. Note that this limit looks exactly as the one in $(1)$; with the difference that in $(1)$ we may choose $\Pi$ depending on $\omega$.
Reference: René L. Schilling/Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Chapter 9.